Let Q be a finite quandle whose right translations commute, with r orbits. Its orbit action determines finite-index lattices H_i in L = Z^r, each containing the respective coordinate vector e_i. We prove that the torsion summand belonging to the ith orbit in second integral rack and normalized quandle homology is T_i = wedge^2 L / (sum_j e_j wedge H_j + wedge^2 H_i). The free ranks are r^2 and r(r-1), respectively. Each orbit size annihilates its torsion summand, and the bound is sharp. A twenty-element example shows that the exponent of the orbit group need not annihilate that summand. Smith matrices with dimensions depending only on the number of orbits compute all invariant factors. The universal coefficient theorem gives second cohomology with any constant abelian coefficient group; in particular, integral second cohomology is torsion-free.

The result is an explicit consequence of established structure-group and covering methods of Darné, Eisermann, and Lebed--Mortier, all credited in the manuscript. It addresses OWR-17294-007 while preserving the original source's unspecified-coefficient cohomology wording. It does not concern arbitrary medial quandles, twisted or nonabelian coefficients, or higher homology.

This six-page English preprint is AI-assisted, self-audited, and unrefereed. Exact checks on 52 finite examples agree with direct boundary-matrix computations. They do not constitute a formal verification of the general theorem. No independent expert review, exhaustive novelty certification, or absolute priority is claimed. PDF, portable LaTeX source, executable checks, and a verification report are provided.
